nLab
trivial torsor

Probably the easiest example of a torsor to understand is the trivial torsor in the topological case.

Definition

Given a space BB and a sheaf of groups, GG on BB, the sheaf of sets underlying GG has a natural left action by GG, which is a sheaf morphism. This is transitive etc. and so gives a torsor, called the trivial GG-torsor, denoted TGT_G.

It is very important to note that TGT_G has TG(B)T_G(B) non-empty (i.e., TGT_G has a ‘global section’), since it is a group so must have an identity element. Conversely any GG-torsor which has such a ‘global section’ is isomorphic to TGT_G.